Problem: Carbon- $14$ is an element which loses exactly half of its mass every $5730$ years. The mass of a sample of carbon- $14$ can be modeled by a function, $M$, which depends on its age, $t$ (in years). We measure that the initial mass of a sample of carbon- $14$ is $741$ grams. Write a function that models the mass of the carbon- $14$ sample remaining $t$ years since the initial measurement. $M(t) = $
The strategy We can model the situation with an exponential function of the general form A ⋅ B f ( t ) A\cdot B\^{ f(t)}, where $A$ is the initial quantity, $B$ is a factor by which the quantity is multiplied over constant time intervals, and $f(t)$ is an expression in terms of $t$ that determines those time intervals. Let's use the given information to determine $A$, $B$, and $f(t)$. Understanding what's given We are given that the initial mass of the sample is $741$ grams, and its half-life is $5730$ years. This means that the initial quantity is $A=741$ and the factor is $B=\dfrac{1}{2}$. We need to find $f(t)$ based on the fact that the quantity is multiplied by $\dfrac{1}{2}$ every $5730$ years. Finding the expression in the exponent We know that the mass of the sample is multiplied by $\dfrac{1}{2}$ every $5730$ years. This means that each time $t$ increases by $5730$, $f(t)$ increases by $1$. Therefore, $f(t)$ is a linear function whose slope is $\dfrac{1}{5730}$. When the initial measurement is made, the sample has all of its mass remaining. So $M(0) = 741$, which means that $f(0)=0$. [Why?] Therefore, $f(t)$ must be $\dfrac{t}{5730}$. Summary We found that the following function models the mass of the carbon- $14$ sample remaining $t$ years since the initial measurement. M ( t ) = 741 ⋅ ( 1 2 ) t 5730 M(t)=741\cdot \left(\dfrac{1}{2}\right)\^{ \frac{t}{5730}}